Weed 1 is Equisetum arvense.
Hard to say what is the second one from that picture.

I don't recognize the second one either, but if it pulls easily and has very shallow roots, it probably is an annual which you can eliminate with a string trimmer. The horsetail is difficult to control with either a string trimmer or by hand pulling.

Thank you for the replies.
I will retake the 2nd weed picture and repost.
Botanybob...do you mean that I would have to resort to chemicals when you say "The horsetail is difficult to control with either a string trimmer or by hand pulling."?

Using a herbicide would be far easier. I think a string trimmer would be almost useless, although if you go after them again and again not letting the green growth exist long enough to replenish the roots, you can exhaust the plant's resources. Hand weeding also is very labor intensive, but perhaps can be done in a shorter time frame. It requires digging up all the underground stems. This probably requires a shovel or spading fork rather than a trowel or weeding tool. With either method, it will require some persistance, and if you turn your back too long, it will reestablish.

There may be some better non-chemical methods too, perhaps smothering with cardboard or plastic, that will eliminate them from an area. I hope someone else will suggest something that has worked for them.

Equisetum (Field Horsetail) is resistant to many herbacides. I would check with your local weed board for what is the preferred non-restricted (home use) chemical in your area. Digging them may only produce new plants if even the smallest pieces of the rhizome are left behind.

Also understand that there are 2 different plant forms of the same plant....one like you see (which is steril) and the other which produces just a spore head and is not branched and steril. The un-branched comes before the branched and once it starts to mature and die back the one you are looking at now comes up. The Equisetums are not true plants but are Sporophytes (ferns) with seriously extensive root systems, hence the reason why many broadleaf selective herbacided don't work. The plants are native to Ontario and other north American locations including here in WA. Many species of wildlife feed on them and they are only a 'weed' where not wanted. Another Equisetum you may be familiar with is Smooth Scouring Rush E. hyemale.
My Weed Science instructor stated that a tea could be made from this to control slugs and was also useful as a pesticide, but I don't have all the details on that :)

Vera

...errr I meant to say the 2nd shoots to appear (branched) are the steril vegatative stage. The smooth unbranched part which comes first is the fertile spore-bearing portion of the plant.

Second weed may be Enchanter's Nightshade. It makes a good 12" high ground cover and will eventually have small white flowers. It has long white roots and is easy to pull out.

The weed whacker will keep them short but will not remove them so you will need to rewhack weekly. Your call as to which is easier -- one time pulling or constant whacking.

I'd really like to know the identity of that second weed. It showed up on our property just a few years ago, and we have thousands of young seedlings this year. I'm ripping them out as fast as I can. It has a lateral kind of root, right? We're finally getting ahead of the garlic mustard and now this!
Let me know if you figure out what it is!

I found this ag web site with some photos.
Do you suppose that second weed is "smooth pigweed"?
I'm trying to compare the pics here....

Here is a link that might be useful: Smooth pigweed

Could it be a pokeweed seedling?

Link: http://www.extension.iastate.edu/Publications/PM746.pdf